Solutions of $p!q! = r!$

Question

The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there?

I came across this about ten years ago; as far as I can tell, it hasn't appeared here before, so I thought that it might be of interest. I'm actually most interested in finding whether there was any progress made since Florian Luca's 2007 article.

Answer 1

The only citation of the Luca paper found by MathSciNet:

Bhat, K. G.; Ramachandra, K.: A remark on factorials that are products of factorials. (Russian. Russian summary) Mat. Zametki 88 (2010), no. 3, 350–354; translation in Math. Notes 88 (2010), no. 3–4, 317–320